50 1 Introduction and Examples
1. Northam Airlines is trying to decide how to partition a new plane for its
Chicago–Detroit route. The plane can seat 200 economy class passengers. A
section can be partitioned off for first class seats but each of these seats takes
the space of 2 economy class seats. A business class section can also be in-
cluded, but each of these seats takes as much space as 1.5 economy class seats.
The profit on a first class ticket is, however, three times the profit of an economy
ticket. A business class ticket has a profit of two times an economy ticket’s
profit. Once the plane is partitioned into these seating classes, it cannot be
changed. Northam knows, however, that the plane will not always be full in
each section. They have decided that three scenarios will occur with about the
same frequency: (1) weekday morning and evening traffic, (2) weekend traffic,
and (3) weekday midday traffic. Under Scenario 1, they think they can sell as
many as 20 first class tickets, 50 business class tickets, and 200 economy tick-
ets. Under Scenario 2, these figures are 10 , 25 , and 175 . Under Scenario 3,
they are 5 , 10 , and 150 . You can assume they cannot sell more tickets than
seats in each of the sections. (In reality, the company may allow overbooking,
but then it faces the problem of passengers with reservations who do not appear
for the flight (no-shows). The problem of determining how many passengers to
accept is part of the field called yield management or revenue management.For
one approach to this problem, see Brumelle and McGill [1993]. This subject is
explored further in Exercise 1 of Section 2.7.)
2. Tomatoes Inc. (TI) produces tomato paste, ketchup, and salsa from four re-
sources: labor, tomatoes, sugar, and spices. Each box of the tomato paste re-
quires 0.5 labor hours, 1.0 crate of tomatoes, no sugar, and 0.25 can of spice. A
ketchup box requires 0.8 labor hours, 0.5 crate of tomatoes, 0.5 sacks of sugar,
and 1.0 can of spice. A salsa box requires 1.0 labor hour, 0.5 crate of tomatoes,
1.0 sack of sugar, and 3.0 cans of spice.
The company is deciding production for the next three periods. It is restricted
to using 200 hours of labor, 250 crates of tomatoes, 300 sacks of sugar, and 100
cans of spices in each period at regular rates. The company can, however, pay
for additional resources at a cost of 2.0 per labor hour, 0.5 per tomato crate,
1.0 per sugar sack, and 1.0 per spice can. The regular production costs for each
product are 1.0 for tomato paste, 1.5 for ketchup, and 2.5 for salsa.
Demand is not known with certainty until after the products are made in each
period. TI forecasts that in each period two possibilities are equally likely, cor-
responding to a good or bad economy. In the good case, 200 boxes of tomato
paste, 40 boxes of ketchup, and 20 boxes of salsa can be sold. In the bad case,
these values are reduced to 100 , 30 , and 5 , respectively. Any surplus produc-
tion is stored at costs of 0.5, 0.25 , and 0.2 per box for tomato paste, ketchup,
and salsa, respectively. TI also considers unmet demand important and assigns
costs of 2.0, 3.0, and 6.0 per box for tomato paste, ketchup, and salsa, re-
spectively, for any demand that is not met in each period.
3. The Clear Lake Dam controls the water level in Clear Lake, a well-known resort
in Dreamland. The Dam Commission is trying to decide how much water to re-
lease in each of the next four months. The Lake is currently 150 mm below flood